A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property (sometimes the condition that κ is regular and uncountable is included).
-Aronszajn trees is undecidable in ZFC: more precisely, the continuum hypothesis implies the existence of an
-Aronszajn tree, and Mitchell and Silver showed that it is consistent (relative to the existence of a weakly compact cardinal) that no
Jensen proved that V = L implies that there is a κ-Aronszajn tree (in fact a κ-Suslin tree) for every infinite successor cardinal κ. Cummings & Foreman (1998) showed (using a large cardinal axiom) that it is consistent that no
If κ is weakly compact then no κ-Aronszajn trees exist.
) implies that all Aronszajn trees are special, a proposition sometimes abbreviated by EATS.
The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there is a club set of levels such that the restrictions of the trees to this set of levels are isomorphic, which says that in some sense any two Aronszajn trees are essentially isomorphic (Abraham & Shelah 1985).
On the other hand, it is consistent that non-special Aronszajn trees exist, and this is also consistent with the generalized continuum hypothesis plus Suslin's hypothesis (Schlindwein 1994).
The special Aronszajn tree T is the union of the sets Uα for all countable α.
We construct the countable levels Uα by transfinite induction on α as follows starting with the empty set as U0: The function f(x) = sup x is rational or −∞, and has the property that if x < y then f(x) < f(y).
This construction can be used to construct κ-Aronszajn trees whenever κ is a successor of a regular cardinal and the generalized continuum hypothesis holds, by replacing the rational numbers by a more general η set.